Riemann Surfaces

Why is ln z "multivalued"? Because the complex plane is not big enough to define it unambiguously. A Riemann surface is the right space where ln z becomes an ordinary single-valued function. This is one of the most beautiful ideas in all of mathematics.

• **String theory:** spacetime compactifications use Riemann surfaces and tori as models for extra dimensions; their topology determines particle physics
• **Algebraic geometry:** smooth algebraic curves over ℂ are precisely compact Riemann surfaces; their classification by genus g is a fundamental problem
• **Cartography:** stereographic projection (the Riemann sphere) is used in polar maps and navigation; conformality means angles are preserved

Предварительные знания

• Analytic Functions
• Analytic Continuation

Multivalued Functions and Branches

A **multivalued function** is a relation that assigns several values to a single argument. Classic example: f(z) = √z has two values for each z ≠ 0. To make the function single-valued, we choose a **branch**: a single-valued analytic selection of the function's values on a connected set.

A **branch** is a single-valued analytic choice of values for a multivalued function on some connected set. For ln z: the principal value Log z = ln|z| + i·Arg(z), where Arg ∈ (-π, π]. A cut along the negative real axis makes Log z analytic.

Branch Points and Cuts

A **branch point** is a point z₀ such that going around it continuously takes one branch into another. At a branch point the function cannot be made single-valued in any neighborhood. To eliminate multivaluedness, one introduces a **branch cut**: a curve from the branch point to ∞.

**Algebraic branch point**: circling n times returns to the original branch (e.g., √z at z = 0). **Logarithmic branch point**: no number of circuits returns to the original branch (e.g., ln z at z = 0). The ramification order is n-1 for z^(1/n).

Riemann Surfaces

A **Riemann surface** is a complex manifold (a generalization of the complex plane) on which a multivalued function becomes single-valued. Instead of branch cuts, one glues together "sheets"-copies of the complex plane corresponding to different branches.

For √z take two sheets ℂ (sheet 0 for the branch e^(iθ/2), sheet 1 for -e^(iθ/2)). The sheets are glued along the cut [0, +∞): the upper edge of the cut on one sheet is connected to the lower edge on the other. Topologically the result is a cylinder.

The Riemann Sphere

The **Riemann sphere** ℂ̂ = ℂ ∪ {∞} is the extended complex plane obtained by adjoining a single point at infinity. It is homeomorphic to the ordinary sphere S² and is a compact complex manifold of dimension 1.

The Riemann sphere is realized via stereographic projection: the sphere S² in ℝ³ is projected onto the plane ℂ from the north pole N = (0,0,1). The north pole corresponds to ∞. Formula: z = (x₁ + ix₂)/(1 - x₃). The projection is conformal: angles are preserved.

Key Ideas

• **Multivalued function**: multiple values for one argument; for single-valuedness choose a branch with a cut
• **Branch point**: circling it takes one branch into another; algebraic (√z) and logarithmic (ln z) types
• **Riemann surface**: the natural domain where a multivalued function becomes single-valued; sheets glued along cuts
• **Riemann sphere** ℂ̂ = ℂ ∪ {∞} - one-point compactification of ℂ, homeomorphic to S²

Related Topics

Riemann surfaces generalize analytic continuation and form the foundation of conformal geometry:

• Analytic Continuation — A Riemann surface is the "maximal domain" of analytic continuation for a multivalued function
• Linear Fractional Transformations — Möbius transformations are precisely the conformal automorphisms of the Riemann sphere
• Riemann Zeta Function — Analytic continuation of ζ(s) beyond Re s > 1 also uses the ideas of Riemann surfaces

Вопросы для размышления

• What does the Riemann surface of z^(2/3) look like? How many sheets does it have and how are they glued?
• Why does the Riemann sphere allow us to speak of "f(∞)" - what does it mean for a meromorphic function?
• Is the Riemann surface of ln z compact? What about the Riemann sphere? What is the fundamental difference?

Связанные уроки

• calc-01-sequences